An intrinsic time to fracture criterion for amorphous polymers

Engineering Fracture Mechanics. 1978. Vol. 10. pp. 659&76. Pergamon Press. Printed in Great Britain

AN

INTRINSIC

CRITERION

FOR

TIME

TO

AMORPHOUS

FRACTURE POLYMERS

K. C. VALANIS and ULKU YILMAZER Division of Materials Engineering,The University of Iowa, Iowa City, IO 52240, U.S.A. Abstract--An energy probability theory of global fracture is formulated using the notion of intrinsic time starting with first principles. Comparisons are made between the theory and experiments on StyreneButadiene Rubber, thus explaining various aspects of failure. It is proposed that fracture occurs as a result of accumulationof broken carbon-carbonbonds. When a critical concentrationof broken bonds is reached, catastrophic failure begins and the specimen fractures. It is assumed that the probability of fracture of a single carbon-carbon bond is determined by the energy content of the bond. Non-uniformdistribution of the free energyof the specimen amongbonds is taken into account by means of an exponential distribution function. The intrinsic time measure pertinent to the constitutive equation of the material is used as the time measure of fracture. This allows use of the time-temperature shift principle, applicable to fracture of polymers, and enables the prediction of lifetimes in high strain rate condtions. Finally the theory predicts correctly that, at constant amplitudes, the number of cycles to failure at low frequencies is directly proportional to the frequency and is independent of the latter at high frequencies.

INTRODUCTION FAILURE of amorphous polymers has been studied under various test conditions by other investigators and theories have been proposed for different aspects of failure[l-7]. However, a theory starting with first principles and explaining all the seemingly different aspects has hitherto been lacking. It will be the purpose of this study to formulate a global fracture theory based on molecular concepts and it will be shown that indeed the theory gives a unified point of view. In this paper essentially four problems, which hitherto remained unrelated, have been given a unified solution. The first three problems concern the determination of the time to break in uniaxial constant strain rate, constant load and constant strain conditions. The fourth concerns the determination of the time to break in high and low frequency cyclic loading conditions as they relate to uniaxial tests. The effect of strain rate is an aspect of fracture which manifests itself in higher frequency cyclic loading as well as in other higher strain rate conditions. Thus a study of cyclic failure has to account for the effect of strain rate to be applicable in the case of higher frequency loading. The comparison of the theory and experimental results is done on SBR, a widely studied amorphous polymer. A final comment is that the interest of this study is in the prediction of lifetimes instead of the stress or strain at fracture, since the knowledge of the lifetime at a particular test condition enables one to determine the stress or strain at break with the use of the constitutive equation of the material in question. PHYSICAL FOUNDATIONS OF THE THEORY In the course of the derivation of the present theoretical failure criterion we consider a macroelement containing N primary carbon-carbon bonds and assume that a critical number N,:r of primary bonds, negligible in comparison to N must fracture before global fracture takes place, i.e. when the specimen fails. This mechanism is supported in the case of crystalline polymers by the experimental findings of Zhurkov and his associates [8-10] who confirmed that a critical concentration of broken primary bonds leads to global fracture. We stipulate that this mechanism is also valid in the case of amorphous polymers as well. The theory begins with a probabilistic view of fracture of a bond--as determined by the energy content of the latter--taking into account the nonuniform distribution of free energy of the specimen among its constituent bonds. It further stipulates, following an earlier paper by Valanis[ll], that fracture initiation takes place on an "intrinsic time scale" which is not absolute, but is a property of the material in question. For the definition and discussion of this concept of intrinsic time the reader is referred to Ref. [12]. 659

660

K.C. VALANISand ULKU YILMAZER THEORY

The failure criterion With the above in mind we consider a macroelement with N number of carbon-carbon bonds having a free energy density of AOr. We define a quantity Aft, the energy stored in a bond

as a4, = 4J- ~R

(1)

where ~b is the current energy and ~bR is the energy of the unstressed, reference state. We also assume that there exists a probability density p'(A~b) such that the probability dp that a bond will break in an increment of intrinsic time dz is [11-12] dp = p'(AO) dz.

(2)

We have assumed that the fracture probability density p'(A~) is determined by the energy content of a bond relative to the unstressed configuration. With reference to the rth bond, eqn (2) is written as dp, = p'(ASr) dz

(3)

and following Ref. [11], the time zf at which the bond will have fractured is given by

~01dpr = SO2/p'(A$,) dz = 1.

(4)

The form of the probability density p'(A~p) can be found in a way analogous to the macroscopic fracture theory [l l], since again the probability of survival of two bonds is the product of the probabilities of their individual survival. This concept leads to the following form of p'(A$): p'(A~b) = 1 - e-C'ta*-*o)

(s)

for A~b> ~bo.In the case of A~b- qJo,

p'(A~) = O.

(6)

In eqn (5) #o is a hypothesized molecular fracture activation energy and c' is a material parameter different from c of Ref.[ll]. The quantity ~b0 should not be mistaken for the dissociation energy in its chemical context. Here we hypothesize that the bond energy is not deterministic, but that its value is probabilistically distributed over a range of values. In this sense ~0 is the lower bound of the distribution. The critical time for bond fracture is given by eqn (7),

fo

Z~(1 - e c'
(7)

with the understanding that (A~,- q~o)= 0 for AqJ-~ ~boand (A~b- ~o) = A ~ - ~o for A~> ~0. On the other hand there exists subcritical times at which there is a finite probability, .less than unity, that a bond will fracture. In particular the time z at which the rth bond will fracture with probability p is given from eqn (8): fo~(1 - e-C'"~*-*O)dz' = p.

(8)

Due to the distribution of energies A~b, there exists at time z a distribution of probabilities p~ The number of bonds Nf that will have fractured in time z, is evidently the sum of all

An intrinsictime to fracturecriterionfor amorphouspolymers

661

probabilities multiplied by the multiplicity nr, i.e.

Nt(z) = ~, nrvr

(9)

r=l or

Nt(z) =

IoZ

nrp'(Af,) dz'.

(I 0)

Let Net be the number of bonds necessary to give rise to global fracture and let the time lapse be z~, In this sense z,,, will be the time at which global fracture will occur. Then z,., is given by the condition

Ncr=

Zcr~=]

fO

n,p'(Af,) dz.

(11)

i.e. Nor = r--~/~"Z nr(l - e -c''a~'r-*°,) dz. dO

(12)

r=l

If the distribution of probabilities is continuous, then

Nr(z) = f o n'p dAf

(13)

where n'(Af) is a probability density function, such that the number of bonds dn between two adjacent states A f and A f + dAft is given by dn

=

n'(Af) dAf.

(14)

Then,

Nt(z ) =

IoZ

n'p'(Af) dAf dz'.

(15)

Then zc, will be given by the condition

Nor = fOzcrf'~ n'p'(hf) d h f dz

(16)

or, substituting for p' from eqn (5),

Ncr= ~0 or

n'(l - e -c'(A*-*cJ)dAf de.

(17)

Equation (17) constitutes on encompassing failure criterion that in principle is applicable to the entire spectrum of deformation histories.

Energy distribution among bonds Although the free energy may be uniformly distributed in the material at the macroscopic level, it will have a probabilistic distribution among the bonds at the molecular level. This may be the result of variations in the length of chains, chain entanglements, local molecular configurations, and the orientation of the macroscopic stress field. For instance, if we examine a hypothetical cube under a uniaxial load, bonds in the chains aligned with the direction of stress will have greater energy than others. EFM Vol, I0, No, 3--N

662

K. C. VALANISand ULKU YILMAZER

The free energy in a chain is a function of chain length between crosslinks. A veriation in the length of the aligned chains will cause an uneven distribution of energy among the bonds in those chains. Bonds on a shorter chain are likely to have greater energies than,the bonds on a longer one. Although the real situation is much more complex, this simplistic model at least gives an insight into the cause of such a distribtuion. Some theories consider the free energy in the material to be uniformly distributed microscopically as well as macroscopically and are oversimplified in this respect. Exceptions, to our knowledge, are Refs. [13-22] where length and stress distributions among chains are considered. Still another aspect of the problem is the variation of strengths of bonds. In reality there is a variation in the strength of primary and secondary bonds. In this work we assume that bonds whose fracture will lead to failure are primary carbon--carbon bonds. We shall concern ourselves with the free energy distribution among such bonds. Evidently the shape of such a distribution is a function of the total energy. In this work we assume that the total free energy A~br with respect to reference state is exponentially distributed among N bonds. This assumption results in the physically sound predictions which we discuss following eqn (23). The exponential distribution is a particular case of the Weibull distribution when the "shape parameter" is equal to unity. A Weibull distribution with the shape parameter less than unity could also represent the situation and, in fact, it gives a better agreement between the theory and the experiment, but in this work we assume that the distribution is exponential for A~ > 0, i.e. the number of bonds dn in an energy interval (A$, Aq, + dAg,) is given by dn = NS' e-S'~* dAO

(18)

where N is the total number of bonds in the macroelement and S' is a constant. This implies that n' = NS' e -s'a*.

(19)

The constraints on the distribution are: f n' dA$ = N

(20)

f o n' A$ dA0 = ASr

(21)

where Atkr is the total free energy in the macroelement. From the last constraint it is found that S'=

N AOr"

(22)

Then, eqn (19) becomes n' - N2 e-ta*Nla*r~ - Aqtr

(23)

The consequences of the last equation are as follows: (i) Equation (23) predicts a large number of unstressed or relatively weakly stressed bonds which would lie on longer chains. The experimental data on polypropylene[23] qualitatively shows this trend. Since the distribution is the result of the configuration of chains it is also expected to be valid for amorphous crossli~ed polymers. (ii) It shows that when the total energy of a specimen increases by its being extended, the

energy will ~ incd~singly more uniformly distributed amoung bonds. Thisis physically-sound since more chains participate in bearing the load when the extension is 8renter. (iii) In the linear elastic range the total free energy A0r is proportional to Young,s modulus

An intrinsictime to fracturecriterionfor amorphouspolymers

663

of elasticity and hence to the crosslink density, therefore for higher crosslink densities eqn (23) shows that the free energy is more uniformly distributed among bonds. Such an outcome is expected, since again more chains bear the load when the crosslink density is greater. (iv) We have assumed that Nor is negligible in comparison to N. This implies that the distribution is unchanged throughout a test if the total free energy A¢JT is constant. However, if the total free energydecreases as a result of relaxation or breakage of primary carbon-carbon bonds, then the load is carried by a smaller number of chains and the distribution becomes more uneven. As an example the failure of one bond in a chain causes dissipation of free energy stored in all others of the same chain and the bonds on other chains carry more of the load thus having greater energies than they had initially. These observations can be predicted by a decrease of the total free energy A~br in eqn (23). Thus we assume that an exponential distribution adequately represents the physical situation and proceed by substituting eqn (23) in eqn (17). We find that

2 e -(a*Nta*r~dAd, Ncr io*" f f (1 - e -~'(a*-*o))N A$r " dz, o =

(24)

because the energy levels between zero and $o do not contribute to the integral. Omitting the algebra and letting c" = (c'/N) and a = N4'0, the result is: Ncr

=N (

Zcr

tt

AI//rC"

e -I'[a*r' dz.

Jo a~bTc"+ I

(25)

It was found that for the values used for c", A~brc" was much less than 1 in the denominator so the fracture criterion used in this-work is

f0

~"Ag, r e-~,/a~,rdz =

Nor

Nc"

(26)

With the above stipulations eqn (25) is a fracture criterion which will predict times to fracture of crosslinked amorphous polymers provided that the deformation history is given and the constitutive dependence of AST on the history is either known or can be calculated.

Rate and nature of damage The rate of damage R is given by the number of bonds breaking in an increment of time, divided by that time increment, i.e. R = dN1 = f ~ (1 - e -c'ta*-~'o)) N"2 e_~a,Nta,r J dA$ dz A~bT

(27)

This can be then written using the same procedure as in the derivation of eqn (25)

A$TNC" e_(,ta,rl.

R-A~"+I

(28)

The last equation shows that damage rate depends on time as a result of the time dependence of ASr and it predicts that, for amorphous crosslinked polymers at their rubbery state, R, at large times, is nearly constant for constant strain or constant load experiments, but is not so for constant strain rate tests. Also, observation of eqn (5) reveals that the bonds having greater energy will have a greater probability of fracture. Damage will occur throughout the specimen as it deforms, as can be observed with EPR techniques. Cracks are likely to initiate at one or more sites. The critical condition at a particular rate is reached according to eqn (26). This condition is tantamount to unstable crack propagation and fracture of the specimen. Equation (28) also shows that damage accumulation is irreversible [22]. Reducing the free

664

K.C. VALANISand ULKU YILMAZER

energy by unloading for example wilt merely lower the rate of damage, but bond rupture will continue at a reduced rate, sometimes to a point where it is not measurable. TIME TO FRACTURE IN UNIAXIAL CONSTANT STRAIN RATE, CONSTANT LOAD AND CONSTANT STRAIN CONDITIONS Following the work at the Natural Rubber Producers' Research Association in England several investigators have done experimental and theoretical studies on polymers to determine the "time to break" under uniaxial test conditions. Among the theories, often referred to, are the works of Greensmith, Bueche and Halpin, Zhurkov and Knauss[l-6]. Although their theories provide good agreement with experimental work, they depend on what we believe to be oversimplified assumptions or have limited applicability. The theories of Bueche-Halpin and Zhurkov are formulated in one dimension in terms of stress. The theory of Zhurkov predicts a finite probability of bond rupture for all values of stress including ¢r = 0; despite this feature it has been found[24] that the theory gives good agreement with experiment for constant stress rate tests in particular. Bueche and Halpin's theory is based on the overrestrictive assumption of constant crack speed during a test, an assumption which is unrealistic unless the free energy in the specimen is constant during the test. The theories of Greensmith and Knauss use the concept of stored energy density and may thus be applied to all test conditions. The time dependence of free energy density may be either introduced in an approximate way by the method of Knauss or, following Greensmith, it may be neglected in the rubbery state, which we are concerned with here. We chose the second option in the applications due to reasons which we will discuss later. Greensmith's theory uses a modified Griffith criterion and depends on the pre-existence of cracks. However, it transpires that the size of the assumed pre-existing cracks is unrealistically high[l]. The characteristic fracture energy T used in the theory was found to be rate dependent, but it was necessary to use different values of parameters for this dependence in order to predict the time to break for cut and uncut speciments. Greensmith's theory makes use of the far field free energy density whereas in Knauss's theory the total energy of the specimen enters his proposed fracture criterion. Also Knauss considers the possible distribution of bond strengths, instead of a more realistic, "distribution among bonds of the free energy of the specimen". In this section we use eqn (26) to predict the failure of amorphous crosslinked polymers in uniaxial constant strain rate, constant load and constant strain condition. We will be interested in their rubbery state, i.e. when their ambient temperature is above their glass transition temperature and the deformation is applied under low strain rate conditions, so that the time-temperature dependent modulus E(t) is at its characteristic rubbery value, for the polymer, at the time of fracture.

Discussion o[ the [allure envelope A great deal of experimental work has been done by T. L. Smith, and a summary of his work can be found in a review by him[25]. A fracture envelope was proposed which states that a plot of stressf to break crop"corrected" by the temperature ratio TolT as predicted by the statistical theory of rubbermvs the strain to break (,~b - 1) forms an envelope for a variety of simple strain histories and ambient temperatures. Specifically, the results of fracture at constant load, constant strain and constant strain rate fall on the same envelope. This envelope is shown in Fig. 1. Essentially it consists of two branches OA and AB. Branch OA, which we shall call the "lower branch", corresponds to strain histories that have been substantially slow in the recent past whereas the "upper branch" AB corresponds to histories with faster strain rates. For the sake of subsequent discussion we shall refer to part A as the "extremum point." There are at least three histories that are exceptions, in the sense that their results do not fall on the envelope. These are discussed below: (I) In constant strain rate tests carried out at "high temperature-low strain rate conditions", tThroughoutthe paper the stress is calculatedper unit undeformedarea.

An intrinsic time for fracture criterion for amorphous polymers

~-STRESS \RELAXATION ~CONSTANT \ \ STRAIN \

u

665

BI ~ - ~ A~ / /

o

Log (Xb-1)

Fig. 1. Schematic representation of failure envelope

which give small orb values, the scatter is very little; it is much greater in "low temperature-high strain rate tests" which give points associated with higher values of ~b; it is particularly bad on the upper branch AB of the fracture envelope. This may reflect the inapplicability of time-temperature superposition for the upper branch as pointed out by T. I_,. Smith [25]. (II) "Dual strain tests" have been done by Knauss. In these, a strain rate R1 is applied to a specimen for a given time period and then another strain rate R2 is applied until the specimen breaks. For a fast-slow sequence with RJR2 = 100 the points fall on the envelope, but for the slow-fast sequence with RI/R2 = 0.01 the points are slightly above the enveh)pe, i.e. ;tb was less than it would be expected from the failure envelope [6]. (III) The results of cyclic tests form another envelope which is slightly below the envelope formed by constant load, constant strain and constant strain rate tests [25]. Smith also observed[25] that the lower branch of failure envelope and the equilibrium stress-strain curves calculated from the theory of Treloar[26] were identical for natural rubber, butyl rubber (sulfur cured), butyl rubber (resin cured), and silicone rubber and not very much different for SBR-I and Viton B vulcanizates. Our understanding of the failure envelope, in the light of the above findings, and the three exceptional cases (which do not obey to the failure envelope rule) is as follows: The test conditions associated with the lower branch OA are such that the material is at or near equilibrium when the point of failure is reached. Figure 2, taken from Ref. [27], shows a series of stress relaxation tests ending with failure, and clarifies our point of view. Note that the curves are horizontal during a substantial part of the lifetime of the specimen. If at failure the rubbery state is reached, i.e. if the material relaxes completely before it fractures, then one would expect the failure points to lie on the equilibrium stress-strain curve.

14 12 ~ : ~ - o 7 5 0 0 % ~'~""'""@475 % 10 450°/°

,

0 x

/

F 400 % 425%

.

~I"E 8 ~ . _

375 %

~t

550%

t = 1545 ®

b 4

®BREAK POINTS (STRESS) % STRAIN t TIME AT BREAK L i ,0 20 310 410 510 610 7O I x 10-2 s e c

8o

: 181 ®

"v-- t = 5 8 8

~-

Fig. 2. Stress relaxation curves ending with failure

666

K C. VALANIS and ULKU YILMAZER

Similarly in creep tests the failure points are very close to equilibrium; constant strain rate test that give points on the lower branch of the envelope indicate that the test conditions are nol very far from equilibrium and hence the end points lie near the equilibrium stress-strain curve. This point of view provides an explanation for the exceptional cases mentioned: (I) In constant strain rate tests, which give points on the upper branch AB, conditions are such that equilibrium is not reached at failure, because either the temperature is too low and/or the strain rate is too high. Therefore, for a given ~b say, the value of Ab as expected from the equilibrium stress-strain curve is not reached and the envelope bends over and gives rise to the upper branch AB. (II) In the case of dual strain rate test with a fast-slow sequence, again the fracture points are such that the material has relaxed almost completely and the experimental points are on the equilibrium stress-strain curve; but in a slow-fast sequence, relaxation is not complete and the points lie above the equilibrium stress-strain sequence. In other words for a given ~b, the value of Ab as expected from the equilibrium stress-strain curve is never reached at the time of failure. Knauss gave an explanation of this phenomenon by saying that constant load, constant strain rate and constant strain tests and the fast-slow sequence dual strain rate tests have strain histories such that d2~/dt2<0, but the slow-fast sequence dual strain rate test has the strain history such that d2¢[dt2>0 and it is the only test which does not lie on the envelope[6J. This strain history criterion associated with the sign of d2~/dt 2 cannot account for the facts mentioned in I. (III) The cyclic test points are actually constitutive points since they represent the cyclic strain response of the material to a cyclic stress input. As such they don't actually belong on the fracture envelope. The fact that they lie near the fracture envelope is evidence of the fact that they were carried out in the vicinity of the equilibrium state. Finally the equilibrium stress-strain curve which is associated with the lower branch of the failure envelope is not obtained experimentally, but is calculated using the Langevin formulation of the kinetic theory of rubber[26]. We accept, however, that the failure envelope represents approximately the equilibrium stress-strain curve until the extremum point A, with less and less accuracy as the fracture strain increase.

Calculation of free energy In the light of the above explanation we assume that tests which give fracture points on the lower branch of the failure envelope are carded out at or near equilibrium conditions; furthermore, the dissipation is small and the equilibrium free energy can be assumed to represent to an adequate degree of accuracy the time dependent free energy[25]. Note that this assumption is only true for the lower branch of the failure envelope and gets worse for the tests the end point of which lies near the extremum point A. We have covered the range up to the extremum point and we denoted this as the "rubbery state". For applications to the entire range the free energy may be approximated as in the theory by Knauss [6]. Observation of Fig. 2 reveals that it is safe to use the equilibrium free energy instead of the time dependent one, since relaxation is negligible during a large period of time compared to duration of the test. In the case of large loads the reduction in lifetime is compensated by an increase in the rate of relaxation so that again we can replace the time dependent free energy with the equilibrium free energy. Thus, the equilibrium free energy can be used under these assumptions for constant strain rate, constant load and constant strain tests. It can be calculated from the area below the equilibrium stress-strain curve (the failure envelope for our purposes). It was found that the log a vs log(A - l) was to a good approximation, a straight line especially for low values of ~r, in which case, or = a(A - 1)b.

(29)

Evidently A~bT can now be obtained readily from eqn (30). A~br = fl Aor dA'

(30)

An intrinsic

time to fracture

criterion

for amorphous

polymers

667

and hence: (31)

(32) Equation (31) was obtained by different arguments in the theory of Greensmith[l].

Endochronic

time measure

In the endochronic theory of fracture [ 1l] the intrinsic time measure is the one used in the formulation of the constitutive equation of the material in question. In the case of amorphous noncrystallizing polymers over a wide range of low strain rates the approximation of dz by dt/gaT is adequate and is indeed the one to which the endochronic time measure reduces at low strain rates. For the entire range the intrinsic time measure was given by [12] dz2 = Pears dC,@d&j + g,2 dt2

(33)

where Pas* and g, are material parameters which may depend on the Right Cauchy-Green tensor Cm@Alternatively (34) It has been shown[28] that the right-hand side of eqn (34) can be expressed in the spatial coordinate system in the form dz ’ dt = Pi&&k/ + g,’

(>

(35)

where dji is the strain rate tensor. For an isotropic material under uniaxial loading, this reduces to

(36) where k, is a material parameter which may depend on the extension ratio through the invariants of dibt Equation (36) can be written as (37) where

At asymptotically low strain rates k12h’2/h2 is much less that l/g* and dz can be approximated by dtlg. For polymers the temperature dependence of dz is given by the timetWe

assume

that both k, and g, are positive.

K. C. VALANISand ULKU YILMAZER

6bg

temperature superposition principle as dz =

dt gaT

(39)

where ar is the temperature shift factor. Time to [racture Constant stress or constant strain tests. Under these test conditions A~Or is assumed to be

independent of time and using eqn (39) eqn (26) can be rewritten as NC r e (alas'for)

t,r = gaT Nc" ACt,

(40)

Equation (40) yields the time to fracture as a function of energy at break A$T,. This equation can be rewritten in terms of tr or a by using eqn (31) and (32), since these equations apply at break also. The values of tr and A at break are denoted as ~rb and Ab respectively. Constant strain rate tests. Under conditions of constant strain rate (A - l) = ~t

(41)

where ~ is the strain rate. Equation (31) can be used to yield: A ~ - = A(i.t) B

(42)

where a

A = -;-7-7.and B = b + 1. o+1

(43)

Substitution of eqn (42) into eqn (26) gives B(Affr~,) 'm

Ncr

t~r = gar Nc" fa*rc, e-~la*r A~b~m dA¢ d 0

(44)

'

Equation (44) is the time to fracture criterion under conditions of constant strain rate.

Comparison with experimental data

The validity of the theoretical fracture criterion given by eqn (26) was tested by comparing the theoretically predicted and the experimentally observed times to fracture in three different types of tests: stress relaxation, creep, and extension at a constant strain rate. As discussed previously we shall limit ourselves to situations where the material spent most of its lifetime near equilibrium. In such cases we can assume that these tests were carried out essentially under equilibrium conditions in which case ACJr is uniquely related to A or o, by means of the expression (31) and (32), where the parameters a and b can be assumed known, since they can be calculated either from the failure envelope or the equilibrium stress-strain curve. This being the case, it is evident from eqns (40) and (44) that once the parameters a and (Nc~gar/Nc") have been determined then the theoretical times to fracture in all three tests can be calculated unambiguously. In the present paper we determine these parameters in constant strain conditions and then proceed to use their values so determined, to predict theoretically the times to fracture in creep and extension under constant strain rate. Greensmith's data [1]. Experimentally observed times to fracture of SBR in conditions of constant strain and constant strain rate histories are shown in Fig. 3, where A0rc, is plotted versus t,r. To determine the constants a and Ncrgar/Nc" two experimental points, t = 1 sec and

An intrinsic time to fracture criterion for amorphous polymers 10

~

f

,

669

i

4 LIJ J D

o

o

o

[]

CONSTANT STRAPN

o

CONSTANT THEORY

-1

O.1

STRABN R A T E

i

I

L

L

1

10

1OO

l,OOO

10,OOO

tc,(Sec) Fig. 3. Energy at break vs time at break

t = 103 sec, approximately, on the constant strain history data, were used. It was found that a = 13.7 j°ule Ncrgar joule-sec ----]' cm Nc" - 0.13 For the purposes of comparing with Greensmith's experimental findings, the constant "a" is not necessary, and B was given as equal to 1.7. These values of the parameters were then used to predict time to fracture in conditions of constant strain and constant strain rate histories. The resulting theoretical curves are shown in Fig. 3. The predictive capability of the theoretical fracture criterion seems quite satisfactory. Halpin's data [4]. Figure 4 shows times to fracture under conditions of constant strain at various stress levels and various ambient temperatures. The temperature effect in that plot has been accounted for by a shift on the time axis in accordance with the time temperature superposition principle using the WLF equation., The parameters a and (Nc,gar/Nc") were determined so as to provide an optimal fit between the theoretical curve and the experimental points, using values of a and b determined in the usual fashion from the fracture envelope given in 2.5

2.0 N

E ~"

1.5

Ooo

v

o ~

oo

1.O (D

S 0.5

THEORY 0 -2

i

I

I

I

I

I

I

-I

0

I

2

3

4

5

Im,o,

Fig. 4. Time to break at constant strain tThis, in fact, is anticipated in the definition of intrinsic time.

67(1

K

C. V A L A N I S

and

ULKU

YILMAZER

~5 ....................................

2.0 [

~

o

1.0 ~ b

0.5 •

-2

THEORY

-1.5

I

I

- I

I

I

I

O

1

2

3

4

5

7

Fig. 5. Timeto breakat constanttoad 2.5

o

7'.O

1.5 o

o

6~

c9 O _J

°°

oO

0.5

' THEORY -2

-I

i

,[

i.

i

I

i

I

.L_

0

1

2

3

4

5

6

7

Fig. 6. Timeto breakat constantstrainrate Ref.[4]. These values were a = 4.44 Kg/cm2,

b = 0.6 at T = - 10°C

The values of a and NcrgarlNc" thus determined were 37.6Kg/cm 2 and 315Kg-mintcm 2, respectively. The times to fracture in conditions of constant strain, constant load and extension under constant strain rate were then calculated and plotted vs ~rcras shown in Figs. 4--6, respectively. Considering the experimental scatter, the agreement between theory and experiment appear quite reasonable. TIME TO I ~ C T U I ~

IN CYCLIC CONDITIONS

Failure under cyclic stress or strain conditions indicates that fracture is a time-dependent process, since the probability of fracture increases as a specimen is cycled. It is observed that the lifetime of a specimen is a functional of the deformation history, in this case the amplitude and the frequency of deformation. In applying the energy probability theory of intrinsic time to fracture in cyclic conditions, the variation of stress or free energy density should be accounted for by properly integrating the appropriate function in question. However it has been found that such a procedure does not lead to the correct prediction of lifetimes for all frequencies, when the intrinsic time is the Newtonian time at some reference temperature. Agreement between the theory and experiments is satisfactory for low frequencies, but gets worse as frequency increases.

An intrinsictimeto fracturecriterionfor amorphouspolymers

671

The same observation applies to the tearing theory of rubbers[I,29] and the theory of Zhurkov [5] in the sense that the theories require some modification to apply in high frequency cyclic conditions. The modifications that have been done [7, 30-32] on the theories mentioned will be discussed later. In a review by Regel and Leksovsky[31] three types of frequency dependence are mentioned for crystalline polymers which seem to contradict each other if the range of frequencies employed in the experiments is not specified. In the first case of low frequencies, the lifetime is independent of frequency. If the time dependence of stress tr or free energy density A~r are properly substituted into the theories of Zhurkov and Greensmith respectively, the theories would need no change. In the case of high frequencies, lifetime is inversely proportional to the frequency. In the third case corresponding to intermediate frequencies the lifetime decreases as the frequency increases, though not yet inversely proportional to the latter. The number of cycles to failure m is by definition proportional to lifetime tcr and frequency f with the realtion m = tcr[,

(45)

so the cases mentioned above yield the following: At low frequencies the number of cycles to failure is directly proportional to the frequency; at high frequencies it is independent of frequency and it is an increasing function of frequency in the intermediate case. The whole spectrum of frequency dependence is observed by Lake and Lindley[7] on SBR by employing frequencies of 0.I-1000 cpm at room temperature. Lake and Lindley [7] modified the theory of tearing of rubbers by assuming that there is a static and a dynamic component to cut growth acting in parallel. However, the assumption of a dynamic component may give rise to a discrepancy between the theory and an experiment of constant strain rate,t since the static component of the theory of Greensmith already gives an adequate explanation of constant strain rate test results[l]. In the modification of the theory of Zhurkov by Bartenev et a1.[32] the decrease in cyclic lifetime with an increase in frequency is attributed to general or local heating of the specimen due to hysteresis. However, it is found[31] that the general temperature rise necessary to observe the reduction in the lifetime is much lower than measured experimentally. Although the local temperature rise cannot be determined precisely, appearance of the same phenomenon in temperature insensitive materials such as natural rubber[7] indicates that local temperature rise cannot account for the reduction in cyclic lifetime for all materials. The discrepancy at high frequencies is also attributed to structural changes being different in cyclic and static loadings and incompleteness of relaxation processes at high frequencies giving rise to high local overstresses and thus making the lifetime shorter[31]. This point of view considers a change in the coefficient 3' of Zhurkov's theory [5] from static to cyclic loading. In our point of view these two effects cannot be separated. The theory presented here considers the free energy distribution among bonds of the specimen and the deformation history as the causes of fracture and includes the aspects mentioned above in the free energy density term. According to our theory the entire history of the specimen should be accounted for if the lifetimes in different loading regimes are to be compared. We will first relate constant strain tests to cyclic tests at a constant frequency for SBR, and then we will find the lifetimes at all frequencies, i.e. we will explain both the amplitude and frequency dependence of lifetimes in cyclic conditions. Application of the theory to cyclic conditions We approximate the cyclic history by a series of loadings and unloadings with the same absolute value of constant strain rate. The variation of strain • with time t is shown in Fig. 7. For highly crosslinked rubbers it can be assumed that the stress strain tr-~ relation is unaltered for all Ioadings and unloadings, i.e. tr is a function of ¢ independent of time. This has tConstantstrainrate test is an extremecase of a cyclictest whenthe lifetimeis a halfperiod.

672

K C. VALANISand ULKU YILMAZER

o Cmax

At

J

2,',t 3,~t 4/,t

6&t

8,',t

Time

Fig. 7. Strainhistoryin a cyclictest been observed for the SBR to be analyzed[7]. Thus eqns (2%32, 43) will be used later for all repetitive cycles. If the absolute value of the strain rate and the strains undergone are the same for all Ioadings and unloadings, the change in intrinsic time Az for each half cycle is the same as implied by eqn (37) and the relation zcr=2mAz

(~)

holds. Furthermore the amount of damage done in each half cycle is the same and the general fracture criterion can be written as 2m

0Az

Scr

ACt e -t"/a~r) dz = Nc""

(47)

Before analyzing the behavior at low and high frequencies, we note that the strain rate in constant strain rate cyclic loading is related to the frequency by

IXI 2fEmax. =

(48)

Thus, limiting cases of low and high frequencies mean limiting cases of low and high strain rates respectively. L o w frequencies. At sufficiently low strain rates the first term at the right hand of eqn (37) is negligible in comparison to llg 2 and dz at constant temperature reduces to dt

dz = -g

(49)

which is the time measure employed in slow noncyclic uniaxial tests, The time to break can be evaluated by 2m

ACt e -'*~A#r)d t - No,

g - Nc'---;"

~O ~d

(50)

The result is: into, 2 m A t = t~, = ~ N c / f

BA-,- lm

.A~" e-¢"v'*T)Aegis>dAOr JO

in which A0rm., is the free energy at the maximum strain e~x of a half cycle.

(51)

An intrinsic time to fracture criterion for amorphous polymers

673

Equation (51) shows that at low frequencies the lifetime is independent of the frequency at a given amplitude, i.e. constant Aerm,x. As a result eqn (45) implies that the ~iumber of cycles is directly proportional to the frequency. These are true for all observable amplitudes. It also implies that lifetimes under cyclic and constant strain rate tests are the same when breaking strain is the same. High frequencies. At sufficiently high strain rates the first term on the right-hand side of eqn (37) becomes much greater than l/g 2 and dz reduces to dz = ~

dt

(52)

dz = ~k~ [del.

(53)

or

The critical number of cycles can be found by

2m

fo ~m'xA~r e -(alike,r) kl;L d~ = NNc" cr

(54)

using eqn (53) in (47). The last equation has the form

Ncr

2mF(ema~) = Nc"

(55)

where F(~ma~) is a function of the amplitude given in eqn (54). The last equation shows that at high frequencies the number of cycles to failure m is independent of frequency, thus lifetime is inversely proportional to the frequency as implied by eqn (45).

Number of cycles to failure for the entire frequency range In this section we analyze the data of Lake and Lindley[7] on SBR. The tests are done on thin strips of rubber where the heat buildup and hysteresis are negligible and the stress-strain curve can be considered unaltered for all but the first few cycles. The stored energy density was expressed in the form of eqn (42) with A = 4.64 ckg2and B = 1.46 in Ref. [7]. Tests were also conducted [7] at stress relaxation conditions which we will use to determine the parameters a and gNc,lNc". Neglecting the damage that occurs in the initial loading, the time to fracture in constant strain conditions is given by eqn (40). Figure 8 shows the experimental points of Lake and Lindley along with the theoretical prediction of eqn (40) with the values of a and gNcdNc" being 28 kg/cm2 and 300 kg-min/cm2 respectively. Now that the parameters a and gNcdNc" are determined, one is left with the determination of relative values of kl and g. The value of klg can be determined by observing the number of cycles to failure at different frequencies. The number of cycles to failure for all frequencies is given by

dz d~ = Nc" No, 2m f l m'~A Or e-<~lA*r)~

(56)

674

K . C . VALAN1S and U L K U Y I L M A Z E R 240 220

o

200

~

180

hi eoryEXperinoen!

160 140 '~- 120 I00 8O 60 40 20 0

o

I I.I

I 12

LogA'~T

o 1 1.4

I 1.3 kg/cm z

Fig. 8. Time to break vs energy at break at constant strain

which is 2m

f%a, Ae s

/ g~ 2i g .,

e-~a'a)

2

,1

-1/2 \

• _gNcr

(57)

using eqns (37), (42) and (48). We define a dimensionless function of frequency F(f) by

P(/) = "(/)

mU)

(58)

where for a given amplitude m (f) is the number of cycles at any frequency and m (1) is the number of cycles at a low frequency which has been taken as 1 cycle/min, here. The theory o f Lake and Lindley gives 4 Gq) A #Jr=.x

where GO~) is a function of frequency. The experimental points for F(f) in Fig. 9 are calculated using the experimental G(f) values of Lake and Lindley and the G(I) value of the smoothed curve reported. The value of k~g can now be calculated comparing the number of cycles at various frequencies and finding F([) using eqn (57). The value of klg was found to be 0.17 rain. to give a family of curves around the experimental F(f) values in Fig. 9. In contrast to the theory of Lake and Lindtey, we have a family of curves around the experimental data instead of a single curve. The scatter in the data seems to be of the order of the difference in the curves corresponding to the upper and lower limits of em~xused in the experiments. Figure 9 shows that F(J) and therefore, m is proportional to the frequency at low frequencies and they are indel~ndent of the frequency at high frequencies. Having determined all the parameters, we use the theory to predict the number of cycles to failure at different amplitudes at the frequency of 100 cycle/rain. Figure 10 is the experimental data of Lake and Lindley and the theoretical prediction using eqn (57) and the values of a, gNc,INc" and klg determined previously, The number of cycles is plotted against the maximum free energy Aq'rs,x corresponding to the maximum strain of a cycle.

An intrinsic time to fracture criterion for amorphous polymers

2,0 1.8 1.6 1.4

(max 0

= .8

Emax= 1.6

o

675

0

o

I~mox= 3.0

~' i.¢

.J

.8

o Experiment --Theory

o

.6! .4 .2

-I

/

I

II

0

I

Log f (cycles/rain)

2

Fig. 9. Dependence of F(f) on frequency

o

7

\

I

o Experiment

~Theory

o~

E 61

8 ~ 8 O o ~ o

4

-

3 o

2

0

i 2

i 6

i 8

i I0

I 12

I 14

L 16

i 18

i 20

2i

2

214

,.~'PTm o x k g / c m 2

Fig. 10, Cycles at break vs the maximum energy at a cycle at I00 cycleslmin

DISCUSSION AND CONCLUSIONS The essential contribution of the paper is the derivation from first principles and from considerations at the molecular level, an "intrinsic" time to fracture criterion which in principle is applicable to the entire spectrum of deformation histories. In particular we have demonstrated that lifetimes in uniaxial constant strain rate, constant strain, constant stress as well as cyclic strain conditions can be predicted by means of one and the same form of the criterion. In particular the times to fracture appropriate to the lower branch of the Smith envelope has been determined and shown to agree well with observation. In the cases considered here any quantitative discrepancies between theory and experiment may have resulted from two assumptions: (i) that equilibrium free energy can be used instead of the time dependent free energy; (ii) that the intrinsic time scale dz can be represented adequately by the expression dt/gaT for low strain rates. The fact that good agreement has been obtained between theory and experiment, particularly for high values of to, means that: (i) the fundamental tenets of the theory are upheld, (ii) that the shape of the probability distribution function p'(A~b) is essentially correct, (iii) that the exponential distribution of energies among the bonds is reasonably close to reality, (iv) that the approximations made in the evaluation of the time dependent free energy and intrinsic time dz are adequate,

676

K. C. VALANIS and ULKU YILMAZER

(v) the full definition of intrinsic time must be used for high strain rate conditions. When condition (v) is met in the analysis of cyclic tests and it is seen that at constant ampiitude to high frequency lifetime is drastically different from the low frequency one and the discrepancy cannot be explained unless the effect of strain rate is accounted for. Although the theory is applied to SBR it is observed that frequency has a similar effect on crystalline polymers as well as other materials. It is expected that if the distribution of free energy among bonds and the intrinsic time measure is properly accounted for, one can explain the frequency dependence of other materials. REFERENCES [1] H. W. Greensmith, Rupture of Rubber--Xl. Tensile rupture and crack growth in a noncrystallizing rubber. J. AppL Polym. Sci. 8, Ill3 0964). [2] F. Bueche, Tensile strength of plastics above the glass temperature. J. AppL Phys. 26(9), 1133 (1955). [3] J. C. Halpin and F. Bueche, Molecular theory for the tensile strength of gum elastomers. J. AppL Phys, 35, 36 (1964). [4] J. C. Halpin, Fracture of amorphous polymeric solids: time to break. J. Appl. Phys. 35, 3133 (1964). [5] S. N. Zhurkov, Kinetic concept of the strength of solids. Int. Z Fracture Mech. 1,311 (1965). [6] W. G. Knauss, The time-dependent fracture of viscoelastic materials. Int. Con[. Fracture, Sendal Japan (1965). [7] G. J. Lake and P. B. Lindley, Cut growth and fatigue of rubbers--lI. Experiments on a noncrystallizing rubber. J. Appl. Polym. Sci. 8, 707 (1964). [8] S. N. Zhurkov, V. S. Kuksenko and A. I. Slutsker, Formation of submicroscopic cracks in polymers under load. Soviet Physics-Solid State 11(2), 238 (1969). [9] S. N. Zhurkov, V. A. Zakrevskii, V. E. Korsukov and V. S. Kuksenko, Mechanism leading to the development of submicroscopic fissures in stressed polymers. Soviet Physics-Solid State 13(7), 1680 (1971). [10] S. N. Zhurkov and V. E. Korsukov, Atomic mechanism of fracture of solid polymers. J. Polym. ScL Phys. 12, 385 (1974). [l 1] K. C. Valanis, An energy-probability theory of fracture--an endochronic theory. J. Mechanique 14, 843 (1975). [12] K. C. Valanis, A theory of viscoplasficity without a yield surface, Part 1: General theory, Part 2: Application to mechanical behaviour of metals. Arch. Mech. 23, 517 (1971). [13] K. L. DeVries, B. A. Lloyd and M. L. Williams, Reaction-rate model for fracture in polymeric fibers. J. AppL Phys. 42, 4644 (1971). [14] B. A. Lloyd, K. L. DeVries, M. L. Williams, Fracture behaviour in Nylon 6 fibers. J. Polym. Sci. A-2, 10, 1415 (1972). [15] K. L. DeVries and R. J. Farris, Strain inhomogeneities, molecular chain scission and stress-deformation in polymers. Int. J. Fracture Mech. 6, 411 (1970). [16] A. Peterlin, Radical formation and fracture of highly drawn crystalline polymers. J. MacromoL Sci. Phys. B6(4), 583 (1972). [17] A. Peterlin, Bond rupture in highly oriented crystalline polymers. J. Poly. Sci. A-2, 7, 1151 (1969). [18] V. P. Tamuzh and P. V. Tikhomirov, Failure of oriented materials under tension. Polym. Mech. 9, 409 (1973L [19] H. H. Kausch, On the relation between the kinetic parameters involved in molecular chain breakage and macroscopic failure. Int. J. Fracture Mech. 6, 301 (1970). [20] H. H. Kausch, Application of electron resonance techniques to high polymer fracture. J. MacromoL Sci. Rer. C-4,

243, (1970). [21] V. P. Tamuzh and P. V. Tikhomirov. Lifetime analysis with allowance for bond stress distribution. Polym. Mech. 9, 300 (1973). [22] A. Peterlin, ESR investigation of chain rupture in strain polymer fibers. Proc. of the 22rid Nobel Symposium, Sodergarn, Lidingo, Sweden (June 1972). [23] D. K. Roylance and K. k DeVries, Determination of atomic stress distribution in oriented polypropylene by infrared spectroscopy. Polym. Lett. 9, 443 (1971) [24] M. L. Williams and K. L. DeVries, Electron paramagnetic resonance measurement of strain rate and cyclic effects on bond rupture. Proc. o/the 5th Int. Congress on Rheotogy, VoL 3, p. 139, Kyoto, (1960), University of Tokyo Press (1968). [25] T. L. Smith, Strength and extensibility of elastomer. In Rheology, Theory and Application (Ed. F. R. Eirich), Vol. 5, p. 127. Academic Press, New York (1969). [26] R. G. Treloar, The Physics o[ Rubber Elasticity, 2nd Ed. Oxford University Press, Clarendon, London and New York (1958). [27] T. L. Smith and P. J. Stedry, Time and temperature dependence of the ultimate properties of an SBR rubber a~ constant elongations. J. Appt. Phys. 31, 1892 (1960). [28] K. V. Valanis, Proper tensorial formulation of the internal variable theory. The Endochroaic time spectrum. Arch. Mech. 29, 173 (1977). [29] A. G. Thomas, Rupture of rubber--V: Cut growth in natural rubber vulcanizates. J. Polym. ScL 31, 467 (1958). [30] V. R. Regel and A. M. Leksovsky, A study of fatigue within the framework of the kinetic concept of fracture. Int. Z Fracture Mech. 3, 99 (1967). [31] V. R. Regel and A. M Leksovsky, Polymer fatigue from the standpoint of the kinetic theory of fracture. Polym. Mech. 4, 58 (1969). [32] G. M. Bartenev and Y. S. Zuyev, Strength and Failure of Viscoelastic Materials. Pergamon Press, Oxford (1968).

(Received I0 January 1977)